I'm not sure why you think any digital device shouldn't know what time it is. It's not leaking any kind of personal information, just literal facts about reality.

Maintaining a shared "now" is actually an interesting problem from a relativistic point of view, considering you need time to communicate what time it is. NTP is a relatively simple protocol with some clever tricks around latency; it is organized by strata which go from very precise, authoritative sources (these are atomic clocks at universities, not the NSA) to various levels of "mirrors", down to within your LAN. It is massively distributed and decentralized by nature, to be able to handle everyone to be in sync without overwhelming a handful of primary clocks.

The end device does not need to be able to talk to the internet at all, just to your router (or designated NTP server if you're into that). It is such an old protocol that it is embedded in most consumer routers, and getting a server running in Linux is literally just install, start. You don't need to connect upstream at all, you can absolutely say "on this network I am the god-clock".